Manuscript XX-A@1 · XX-A.tex · section · 3a586110

Raw coefficient calculation

Complete raw and relational lapse through second order › Raw coefficient calculation · lines 594–644

XX-A@1#eq:W0, XX-A@1#eq:W1, XX-A@1#eq:W2, XX-A@1#eq:N0, XX-A@1#eq:N1, XX-A@1#eq:N2, XX-A@1#eq:normalization

Raise four-dimensional indices using $g ˉ$ , and put $v_{μ} = \partial_{μ} ϕ ˉ$ . The inverse metric coefficients are

g_{s}^{μν} = g ˉ^{μν} - s h^{μν} + \frac{s ^{2}}{2} (2 h^{μ}_{ρ} h^{ρ ν} - k^{μν}) .

Let

W_{s} = - g_{s}^{- 1} (d ϕ_{s}, d ϕ_{s}) = W_{0} + s W_{1} + \frac{s ^{2}}{2} W_{2} .

Its coefficients are

W_{0} W_{1} W_{2} = \frac{A ^{2}}{t ^{2}}, = h^{μν} v_{μ} v_{ν} - 2 g ˉ^{μν} v_{μ} \partial_{ν} φ, = (k^{μν} - 2 h^{μ}_{ρ} h^{ρ ν}) v_{μ} v_{ν} + 4 h^{μν} v_{μ} \partial_{ν} φ - 2 g ˉ^{μν} \partial_{μ} φ \partial_{ν} φ - 2 g ˉ^{μν} v_{μ} \partial_{ν} ψ .

For $N_{s} = N_{0} + s N_{1} + s^{2} N_{2} /2$ ,

N_{0} N_{1} N_{2} = W_{0}^{- 1/2} = \frac{t}{A}, = - \frac{1}{2} W_{0}^{- 3/2} W_{1}, = \frac{3}{4} W_{0}^{- 5/2} W_{1}^{2} - \frac{1}{2} W_{0}^{- 3/2} W_{2} .

These follow from two derivatives of $w^{- 1/2}$ .

The cosmic-clock raw lapse is a different normalization:

L_{s} = (- g_{s}^{- 1} (d T_{s}, d T_{s}))^{- 1/2} .

Since $F^{'} (ϕ_{s}) = T_{s} / A$ ,

N_{s} = \frac{T _{s}}{A} L_{s} .

Neither normalization is silently substituted for the other.

Displayed formulas in this passage

XX-A@1#eq:W0 · line 611
$W_{0} W_{1} W_{2} = \frac{A ^{2}}{t ^{2}}, = h^{μν} v_{μ} v_{ν} - 2 g ˉ^{μν} v_{μ} \partial_{ν} φ, = (k^{μν} - 2 h^{μ}_{ρ} h^{ρ ν}) v_{μ} v_{ν} + 4 h^{μν} v_{μ} \partial_{ν} φ - 2 g ˉ^{μν} \partial_{μ} φ \partial_{ν} φ - 2 g ˉ^{μν} v_{μ} \partial_{ν} ψ .$
XX-A@1#eq:W1 · line 611
$W_{0} W_{1} W_{2} = \frac{A ^{2}}{t ^{2}}, = h^{μν} v_{μ} v_{ν} - 2 g ˉ^{μν} v_{μ} \partial_{ν} φ, = (k^{μν} - 2 h^{μ}_{ρ} h^{ρ ν}) v_{μ} v_{ν} + 4 h^{μν} v_{μ} \partial_{ν} φ - 2 g ˉ^{μν} \partial_{μ} φ \partial_{ν} φ - 2 g ˉ^{μν} v_{μ} \partial_{ν} ψ .$
XX-A@1#eq:W2 · line 611
$W_{0} W_{1} W_{2} = \frac{A ^{2}}{t ^{2}}, = h^{μν} v_{μ} v_{ν} - 2 g ˉ^{μν} v_{μ} \partial_{ν} φ, = (k^{μν} - 2 h^{μ}_{ρ} h^{ρ ν}) v_{μ} v_{ν} + 4 h^{μν} v_{μ} \partial_{ν} φ - 2 g ˉ^{μν} \partial_{μ} φ \partial_{ν} φ - 2 g ˉ^{μν} v_{μ} \partial_{ν} ψ .$
XX-A@1#eq:N0 · line 625
$N_{0} N_{1} N_{2} = W_{0}^{- 1/2} = \frac{t}{A}, = - \frac{1}{2} W_{0}^{- 3/2} W_{1}, = \frac{3}{4} W_{0}^{- 5/2} W_{1}^{2} - \frac{1}{2} W_{0}^{- 3/2} W_{2} .$
XX-A@1#eq:N1 · line 625
$N_{0} N_{1} N_{2} = W_{0}^{- 1/2} = \frac{t}{A}, = - \frac{1}{2} W_{0}^{- 3/2} W_{1}, = \frac{3}{4} W_{0}^{- 5/2} W_{1}^{2} - \frac{1}{2} W_{0}^{- 3/2} W_{2} .$
XX-A@1#eq:N2 · line 625
$N_{0} N_{1} N_{2} = W_{0}^{- 1/2} = \frac{t}{A}, = - \frac{1}{2} W_{0}^{- 3/2} W_{1}, = \frac{3}{4} W_{0}^{- 5/2} W_{1}^{2} - \frac{1}{2} W_{0}^{- 3/2} W_{2} .$
XX-A@1#eq:normalization · line 639
$N_{s} = \frac{T _{s}}{A} L_{s} .$
unlabeled · line 599
$g_{s}^{μν} = g ˉ^{μν} - s h^{μν} + \frac{s ^{2}}{2} (2 h^{μ}_{ρ} h^{ρ ν} - k^{μν}) .$
unlabeled · line 606
$W_{s} = - g_{s}^{- 1} (d ϕ_{s}, d ϕ_{s}) = W_{0} + s W_{1} + \frac{s ^{2}}{2} W_{2} .$
unlabeled · line 635
$L_{s} = (- g_{s}^{- 1} (d T_{s}, d T_{s}))^{- 1/2} .$